Optimal paths and the prehistory problem for large fluctuations in noise-driven systems

Dykman, M. I.; McClintock, P. V. E.; Smelyanski, V. N.; Stein, N. D.; Stocks, Nigel G.

doi:10.1103/physrevlett.68.2718

Cited by 145 publications

(189 citation statements)

References 16 publications

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“…When noise is present in the system, a rare, large fluctuation may drag the system away from an initial stationary state to the basin of attraction of the opposite mode. It is known from the theory of stochastic transitions in nonlinear systems that in the limit of small noise strength a transition takes place in a ballistic way along a most probable escape path (MPEP) which can be calculated by solving an auxiliary Hamiltonian system [17]. As our arguments rely only on the topological features of the system, an exact calculation of the MPEP is not required.…”

Section: (G)-(i) Four Stationary Solutions Exist For Eqs (3)-(4)mentioning

confidence: 99%

Topological Insight into the Non-Arrhenius Mode Hopping of Semiconductor Ring Lasers

et al. 2008

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We investigate both theoretically and experimentally the stochastic switching between two counter-propagating lasing modes of a semiconductor ring laser. Experimentally, the residence time distribution cannot be described by a simple one parameter Arrhenius exponential law and reveals the presence of two different mode-hop scenarios with distinct time scales. In order to elucidate the origin of these two time scales, we propose a topological approach based on a two-dimensional dynamical system. 42.55.Px,42.60.Mi Fluctuations in active optical systems such as lasers is one of today's technological challenges as well as a fundamental problem of modern physics as they are the result of the quantum nature of the interaction between light and matter [1]. Fluctuations are e.g. responsible for longitudinal mode switching in semiconductor lasers [2], polarization mode-hopping in Vertical Cavity Surface Emitting Lasers (VCSELs) [3][4][5], and they play a fundamental role in stochastic and coherence resonances of optical systems [6][7][8].Semiconductor ring lasers (SRLs) are a particular class of lasers whose operation is strongly affected by stochastic fluctuations. The circular geometry of the active cavity allows a SRL to operate in two possible directions, namely clockwise mode (CW ) and counter-clockwise mode (CCW ). From the application point of view, SRLs are ideal candidates for all-optical information-storage. [9][10][11]. From a theoretical point of view, SRLs represent the optical prototype of nonlinear Z 2 -symmetric systems [12], which appear in many fields of physics.Fluctuations induce spontaneous abrupt changes in the SRL's directional operation from CW to CCW and vice versa, and therefore represent a major limitation to their successful applications for instance as optical memories. An in-depth understanding of the mode-hopping in SRLs would shed light on the stochastic properties of the large class of Z 2 -symmetric systems. In spite of its importance, the problem of spontaneous directional switches in SRLs remains unaddressed, partly due to the high dimensionality of the models that have been proposed for SRLs [9,13].In this paper, we address the problem of such fluctuations both theoretically and experimentally. We experimentally investigate the properties of the residence time distribution (RTD) that quantifies the mode hopping. Our theoretical analysis is based on an asymptotic reduction of a full rate-equation model to a Z 2 -symmetric planar system [14].We consider here an InP-based multiquantum-well SRL with a racetrack geometry and a free-spectralrange of 53.6 GHz. The device operates in a singletransverse, single-longitudinal mode regime at wavelength λ = 1.56µm. However, it will be clear from the rest of the discussion that our analysis is general and applies to any kind of ring geometry. A waveguide has been integrated on the same chip in order to couple power out from the ring. This bus waveguide can be independently biased in order to reduce absorption losses. The waveguide crosses the fa...

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Section: (G)-(i) Four Stationary Solutions Exist For Eqs (3)-(4)mentioning

confidence: 99%

Topological Insight into the Non-Arrhenius Mode Hopping of Semiconductor Ring Lasers

et al. 2008

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“…As discussed above, in any diffusion step the mover is initially set to the crossing point of the next plane along the global search path and the vector r ជ final i Ϫr ជ current i . Therefore, the mover will be always focused back onto the global search direction so that the search path cannot diverge which is the major difficulty in another study 5 …”

Section: Finding True Diffusion Barriers By the Projected Conjugate Gmentioning

confidence: 99%

Locally activated Monte Carlo method for long-time-scale simulations

et al. 2000

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We present a technique for the structural optimization of atom models to study long time relaxation processes involving different time scales. The method takes advantage of the benefits of both the kinetic Monte Carlo ͑KMC͒ and the molecular dynamics simulation techniques. In contrast to ordinary KMC, our method allows for an estimation of a true lower limit for the time scale of a relaxation process. The scheme is fairly general in that neither the typical pathways nor the typical metastable states need to be known prior to the simulation. It is independent of the lattice type and the potential which describes the atomic interactions. It is adopted to study systems with structural and/or chemical inhomogeneity which makes it particularly useful for studying growth and diffusion processes in a variety of physical systems, including crystalline bulk, amorphous systems, surfaces with adsorbates, fluids, and interfaces. As a simple illustration we apply the locally activated Monte Carlo to study hydrogen diffusion in diamond.

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“…In particular, it has been unclear whether or not there exists a unique optimal path along which escape from a chaotic attractor takes place. Theoretical predictions of the character of the optimal path distribution near a chaotic attractor do not yet exist.It has been established that fluctuational dynamics can be investigated directly through measurements of the so-called prehistory probability distribution of fluctuations [5,6], making it possible to examine situations for which the use of analytic methods still remains problematic. We have applied this technique to experimental investigations of noise-induced escape from a quasi-attractor and from the Lorenz attractor.…”

mentioning

confidence: 99%

“…It has been established that fluctuational dynamics can be investigated directly through measurements of the so-called prehistory probability distribution of fluctuations [5,6], making it possible to examine situations for which the use of analytic methods still remains problematic. We have applied this technique to experimental investigations of noise-induced escape from a quasi-attractor and from the Lorenz attractor.…”

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confidence: 99%

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Noise induced escape from different types of chaotic attractor

Khovanov¹,

Анищенко²,

Luchinsky³

et al. 2000

AIP Conference Proceedings

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Abstract. Noise-induced escape from a quasi-attractor, and from the Lorenz attractor with non-fractal boundaries, are compared through measurements of optimal paths. It has been found that, for both types of attractor, there exists a most probable (optimal) escape trajectory, the prehistory of the escape being defined by the structure of the chaotic attractor. For a quasi-attractor the escape process is realized via several steps, which include transitions between low-period saddle cycles co-existing in the system phase space. The prehistory of escape from the Lorenz attractor is defined by stable and unstable manifolds of the saddle center point, and the escape itself consists of crossing the saddle cycle surrounding one of the stable point-attractors.A major unsolved problem in the theory of fluctuations is that of noise-induced escape from a chaotic attractor [1]. Chaotic systems are widespread in nature, and the study of their dynamics in the presence of fluctuations is both of fundamental interest, and also of importance in relation to a range of applications, e.g. to stabilization of the voltage standard [2], neuron dynamics [3], and laser systems [4].The difficulty of solving the fluctuational escape problem stems largely from the fact that the dynamics of the system during large noise-induced deviations from deterministic chaotic trajectories remains obscure. In particular, it has been unclear whether or not there exists a unique optimal path along which escape from a chaotic attractor takes place. Theoretical predictions of the character of the optimal path distribution near a chaotic attractor do not yet exist.It has been established that fluctuational dynamics can be investigated directly through measurements of the so-called prehistory probability distribution of fluctuations [5,6], making it possible to examine situations for which the use of analytic methods still remains problematic. We have applied this technique to experimental investigations of noise-induced escape from a quasi-attractor and from the Lorenz attractor. CP502, Stochastic and Chaotic Dynamics in the Lakes: STOCHAOS, edited by

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Optimal paths and the prehistory problem for large fluctuations in noise-driven systems

Cited by 145 publications

References 16 publications

Topological Insight into the Non-Arrhenius Mode Hopping of Semiconductor Ring Lasers

Topological Insight into the Non-Arrhenius Mode Hopping of Semiconductor Ring Lasers

Locally activated Monte Carlo method for long-time-scale simulations

Noise induced escape from different types of chaotic attractor

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